LAUSR

In this paper, we describe completely the ????-singular subgroup of an abelian group and a ????-nonsingular abelian group in terms of the basic subgroups of its p-components and the quotient group by the torsion part. We also prove that a pure subgroup and a quotient group by a pure subgroup of a ????-nonsingular abelian group are ????-nonsingular and give a condition under which a pure extension of a ????-nonsingular abelian group by a ????-nonsingular group is ????-nonsingular.